Hilbert schemes, which parametrize subschemes in algebraic varieties,
have been extensively studied in algebraic geometry for the last 50
years. The most interesting class of Hilbert schemes are schemes
\(X^{[n]}\) of collections of \(n\) points
(zero-dimensional subschemes) in a smooth algebraic surface
\(X\). Schemes \(X^{[n]}\) turn out to be closely related
to many areas of mathematics, such as algebraic combinatorics,
integrable systems, representation theory, and mathematical physics,
among others.

This book surveys recent developments of the theory of Hilbert
schemes of points on complex surfaces and its interplay with infinite
dimensional Lie algebras. It starts with the basics of Hilbert schemes
of points and presents in detail an example of Hilbert schemes of
points on the projective plane. Then the author turns to the study of
cohomology of \(X^{[n]}\), including the construction of the action of
infinite dimensional Lie algebras on this cohomology, the ring
structure of cohomology, equivariant cohomology of \(X^{[n]}\) and the
Gromov–Witten correspondence. The last part of the book presents
results about quantum cohomology of \(X^{[n]}\) and related questions.

The book is of interest to graduate students and researchers in
algebraic geometry, representation theory, combinatorics, topology,
number theory, and theoretical physics.